MULTISCALE COMPUTATIONAL METHODS

MULTISCALE COMPUTATIONAL

METHODS

Achi BrandtThe Weizmann Institute of ScienceUCLA

www.wisdom.weizmann.ac.il/~achi

0u 1u2u

3u4u

given

02

043214

Fh

uuuuu

Poisson equation:2 2

2 2

u uF

x y

Approximating Poisson equation:

given

Solving PDE: Influence of pointwiserelaxation on the error

Error of initial guess Error after 5 relaxation sweeps

Error after 10 relaxations Error after 15 relaxations

Fast error smoothingslow solution

When relaxation slows down:

the error is a sum of low eigen-vectors

ELLIPTIC PDE'S (e.g., Poisson equation)

the error is smooth

The error can be approximated on a coarser grid

LU=F

h

2h

4h

LhUh=Fh

L2hU2h=F2h

L4hU4h=F4h

hLhUh = Fh

LocalRelaxation

hu~approximation

hV hh u~U smooth

2h

4h

L2hU2h = F2h

L4hV4h = R4h

L2hV2h = R2h R2h = ( Fh -Lh )hh2 hu~

interpolation (order l+p)to a new grid

interpolation (order m)of corrections

relaxation sweeps

algebraic error< truncation error

residual transfer

ν νenough sweepsor direct solver*

2ν

Full MultiGrid (FMG) algorithm

..

.

*

1ν

1ν

1ν

2ν

2ν

2ν

h0

h0/2

h0/4

2h

h

**

1ν

1ν

1ν2ν

*

2ν

2ν

2ν

multigrid cycle V

Multigrid solversCost: 25-100 operations per unknown

• Linear scalar elliptic equation (~1971)


• Linear scalar elliptic equation (~1971)*• Nonlinear• Grid adaptation• General boundaries, BCs*• Discontinuous coefficients• Disordered: coefficients, grid (FE) AMG• Several coupled PDEs* (1980)

• Non-elliptic: high-Reynolds flow• Highly indefinite: waves• Many eigenfunctions (N)• Near zero modes• Gauge topology: Dirac eq.• Inverse problems• Optimal design• Integral equations Full matrix• Statistical mechanics

Massive parallel processing*Rigorous quantitative analysis

(1986)

FAS (1975)

Within one solver

)log(

2

NNO

fuku

(1977,1982)


• Linear scalar elliptic equation (~1971)*• Nonlinear• Grid adaptation• General boundaries, BCs*• Discontinuous coefficients• Disordered: coefficients, grid (FE) AMG• Several coupled PDEs* (1980)

• Non-elliptic: high-Reynolds flow• Highly indefinite: waves• Many eigenfunctions (N)• Near zero modes• Gauge topology: Dirac eq.• Inverse problems• Optimal design• Integral equations• Statistical mechanics


(1986)

FAS (1975)

Within one solver

)log(

2

NNO

fuku

(1977,1982)



relaxation sweeps


residual transfer


**

1ν

1ν

1ν2ν

*

2ν

2ν

2ν


..

.

*

1ν

1ν

1ν

2ν

2ν

2ν

Vcyclemultigrid

h0

h0/2

h0/4

2h

h

h

2h

4h

LhUh = Fh

L4hU4h = F4h

h2

h4

Fine-to-coarse defect correction

L2hV2h = R2h

Full Approximation scheme (FAS):

U2h = + V2h L2hU2h = F2h

LocalRelaxation

hu~approximation

hV hh u~U smooth

hu~hh2


• Linear scalar elliptic equation (~1971)*• Nonlinear• Grid adaptation• General boundaries, BCs*• Discontinuous coefficients• Disordered: coefficients, grid (FE)• Several coupled PDEs*• Non-elliptic: high-Reynolds flow• Highly indefinite: waves• Many eigenfunctions (N)• Near zero modes• Gauge topology: Dirac eq.• Inverse problems• Optimal design• Integral equations• Statistical mechanics


Within one solver



relaxation sweeps


residual transfer


**

1ν

1ν

1ν2ν

*

2ν

2ν

2ν


..

.

*

1ν

1ν

1ν

2ν

2ν

2ν

Vcyclemultigrid

h0

h0/2

h0/4

2h

h

h

2h

4h

LhUh = Fh

L4hU4h = F4h

h2

h4

Fine-to-coarse defect correction

L2hV2h = R2h

Full Approximation scheme (FAS):

U2h = + V2h L2hU2h = F2h

LocalRelaxation

hu~approximation

hV hh u~U smooth

Truncation error estimator

hu~hh2


• Linear scalar elliptic equation (~1971)*• Nonlinear• Grid adaptation• General boundaries, BCs*• Discontinuous coefficients• Disordered: coefficients, grid (FE) AMG• Several coupled PDEs*

(1980)

• Non-elliptic: high-Reynolds flow• Highly indefinite: waves• Many eigenfunctions (N)• Near zero modes• Gauge topology: Dirac eq.• Inverse problems• Optimal design• Integral equations• Statistical mechanics


(1986)

FAS (1975)

Within one solver

)log(

2

NNO

fuku

(1977,1982)

• Same fast solver FMG

Local patches of finer grids

• Each patch may use different coordinate system and anisotropic grid and different

physics; e.g. Atomistic



relaxation sweeps


residual transfer


2ν


..

.

*

1ν

1ν

1ν

2ν

2ν

2ν

h0

h0/2

h0/4

2h

h

**

1ν

1ν

1ν2ν

*

2ν

2ν

2ν

multigrid cycle V

• Same fast solver FMG,


• Each level correct the equations of the next coarser level

• Each patch may use different coordinate system and anisotropic grid

• Same fast solver FMG, FAS

x , y

)(

)(

sy

sx

r , s

Finer level with local coordinate transformation

Boundary or tracked layer

With anisotropic further refinement

• Same fast solver FMG,


• Each level correct the equations of the next coarser level

• Each patch may use different coordinate system and anisotropic grid

“Quasicontiuum” method [B., 1992]

• Each patch may use different coordinate system and anisotropic grid and different

physics; e.g. Atomistic

and differet physics; e.g. atomistic

• Same fast solver FMG, FAS


• Linear scalar elliptic equation (~1971)*• Nonlinear• Grid adaptation• General boundaries, BCs*• Discontinuous coefficients• Disordered: coefficients, grid (FE)• Several coupled PDEs* (1980)• Non-elliptic: high-Reynolds flow• Highly indefinite: waves• Many eigenfunctions (N)• Near zero modes• Gauge topology: Dirac eq.• Inverse problems• Optimal design• Integral equations• Statistical mechanics


(1986)

Stokes

0

0

0

yx

y

x

vu

Pv

Pu

0

0

0

0

0

0

p

v

u

yx

y

x

2 yyxxL

L

det

h-principal LLu f Ldet

0

v

u

Riemann

Cauchy

xy

yx

Compressible Navier-Stokes(on the viscous scale)

uk3

2

Central Cauchy-Riemannh2

Central (Navier-) Stokeshh

Q2

Stokes

2D

0

0

0

yx

y

x

20

p

v

u

StokesNavier

ibleIncompress

2D

Q

0

0

0

yx

y

x

Q

Q

1Q u R

p

v

u

Euler

2D

1

0

00

0

0

0

0

0

0

0

0

0 y

x

y

y

x

x

P

u

P

u

u

u

22

23

au

xu

PPa p2

2

p

v

u


• Linear scalar elliptic equation (~1971)*• Nonlinear• Grid adaptation• General boundaries, BCs*• Discontinuous coefficients• Disordered: coefficients, grid (FE) • Several coupled PDEs* • Non-elliptic: high-Reynolds flow• Highly indefinite: waves• Many eigenfunctions (N)• Near zero modes• Gauge topology: Dirac eq.• Inverse problems• Optimal design• Integral equations• Statistical mechanics



• Linear scalar elliptic equation (~1971)*• Nonlinear• Grid adaptation• General boundaries, BCs*• Discontinuous coefficients• Disordered: coefficients, grid (FE)• Several coupled PDEs*• Non-elliptic: high-Reynolds flow• Highly indefinite: waves• Many eigenfunctions (N)• Near zero modes• Gauge topology: Dirac eq.• Inverse problems• Optimal design• Integral equations• Statistical mechanics


ALGEBRAIC MULTIGRID (AMG) 1982

Ax = b

When relaxation slows down:

DISCRETIZED PDE'S

GENERAL SYSTEMS OF LOCAL EQUATIONS

the error is smooth

Along characteristics

The error can be approximated

by a far fewer degrees

of freedom (coarser grid)

the error is a sum of low eigen-vectors

ELLIPTIC PDE'S

the error is smooth


Coarse variables - a subset

Criterion: Fast convergence of “compatible relaxation”

Ax = b

Relax Ax = 0Keeping coarse variables = 0


Coarse variables - a subset

1. “General” linear systems2. Variety of graph problems

General procedures for deriving:* Interpolations

Ax = b

* Restriction* Coarse-level equations

Generalizations:

Graph problems

Partition: min cut

Clustering bioinformatics

Image segmentation

VLSI placement Routing

Linear arrangement: bandwidth, cutwidth

Graph drawing low dimension embedding

MULTISCALE COMPUTATIONAL METHODS

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